# Properties

 Label 900.6.a.q Level 900 Weight 6 Character orbit 900.a Self dual yes Analytic conductor 144.345 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$144.345437832$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{31})$$ Defining polynomial: $$x^{2} - 31$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 20) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{31}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 11 \beta q^{7} +O(q^{10})$$ $$q + 11 \beta q^{7} + 100 q^{11} + 66 \beta q^{13} + 88 \beta q^{17} -2244 q^{19} + 307 \beta q^{23} + 7854 q^{29} -2144 q^{31} -934 \beta q^{37} + 7414 q^{41} + 1595 \beta q^{43} -847 \beta q^{47} -1803 q^{49} + 2178 \beta q^{53} -25972 q^{59} -3058 q^{61} -5279 \beta q^{67} -37608 q^{71} + 2156 \beta q^{73} + 1100 \beta q^{77} + 79728 q^{79} -1463 \beta q^{83} + 826 q^{89} + 90024 q^{91} + 3376 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 200q^{11} - 4488q^{19} + 15708q^{29} - 4288q^{31} + 14828q^{41} - 3606q^{49} - 51944q^{59} - 6116q^{61} - 75216q^{71} + 159456q^{79} + 1652q^{89} + 180048q^{91} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −5.56776 5.56776
0 0 0 0 0 −122.491 0 0 0
1.2 0 0 0 0 0 122.491 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.6.a.q 2
3.b odd 2 1 100.6.a.d 2
5.b even 2 1 inner 900.6.a.q 2
5.c odd 4 2 180.6.d.b 2
12.b even 2 1 400.6.a.s 2
15.d odd 2 1 100.6.a.d 2
15.e even 4 2 20.6.c.a 2
20.e even 4 2 720.6.f.d 2
60.h even 2 1 400.6.a.s 2
60.l odd 4 2 80.6.c.b 2
120.q odd 4 2 320.6.c.d 2
120.w even 4 2 320.6.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.c.a 2 15.e even 4 2
80.6.c.b 2 60.l odd 4 2
100.6.a.d 2 3.b odd 2 1
100.6.a.d 2 15.d odd 2 1
180.6.d.b 2 5.c odd 4 2
320.6.c.d 2 120.q odd 4 2
320.6.c.e 2 120.w even 4 2
400.6.a.s 2 12.b even 2 1
400.6.a.s 2 60.h even 2 1
720.6.f.d 2 20.e even 4 2
900.6.a.q 2 1.a even 1 1 trivial
900.6.a.q 2 5.b even 2 1 inner

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(900))$$:

 $$T_{7}^{2} - 15004$$ $$T_{11} - 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ $$1 + 18610 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 - 100 T + 161051 T^{2} )^{2}$$
$13$ $$1 + 202442 T^{2} + 137858491849 T^{4}$$
$17$ $$1 + 1879458 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 2244 T + 2476099 T^{2} )^{2}$$
$23$ $$1 + 1185810 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 7854 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 2144 T + 28629151 T^{2} )^{2}$$
$37$ $$1 + 30515770 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 7414 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 21442214 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 + 369731298 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 + 248174170 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 25972 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 3058 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 755362070 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 37608 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 + 3569749522 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 79728 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 + 7612675530 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 826 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 + 15761405890 T^{2} + 73742412689492826049 T^{4}$$